Latin hypercubes realizing integer partitions

Abstract

For an integer partition $h_1+\dots +h_n=N$, a 2-realization of this partition is a latin square of order N with disjoint subsquares of orders $h_1,\dots ,h_n$. The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to m-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.